Previous |  Up |  Next


Emden-Fowler equation; group invariant solution; least energy solution; positive solution; variational method
We study the existence of positive solutions for the $p$-Laplace Emden-Fowler equation. Let $H$ and $G$ be closed subgroups of the orthogonal group $O(N)$ such that $H \varsubsetneq G \subset O(N)$. We denote the orbit of $G$ through $x\in \mathbb {R}^N$ by $G(x)$, i.e., $G(x):=\{gx\colon g\in G \}$. We prove that if $H(x)\varsubsetneq G(x)$ for all $x\in \overline {\Omega }$ and the first eigenvalue of the $p$-Laplacian is large enough, then no $H$ invariant least energy solution is $G$ invariant. Here an $H$ invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all $H$ invariant functions. Therefore there exists an $H$ invariant $G$ non-invariant positive solution.
[1] Bhattacharya, T.: A proof of the Faber-Krahn inequality for the first eigenvalue of the $p$-Laplacian. Ann. Mat. Pura Appl., IV. Ser. 177 (1999), 225-240. DOI 10.1007/BF02505910 | MR 1747632 | Zbl 0966.35091
[2] Borel, A.: Le plan projectif des octaves et les sphères comme espaces homogènes. French C. R. Acad. Sci., Paris 230 (1950), 1378-1380. MR 0034768 | Zbl 0041.52203
[3] Kajikiya, R.: Least energy solutions of the Emden-Fowler equation in hollow thin symmetric domains. J. Math. Anal. Appl. 406 (2013), 277-286. DOI 10.1016/j.jmaa.2013.04.068 | MR 3062420
[4] Kajikiya, R.: Multiple positive solutions of the Emden-Fowler equation in hollow thin symmetric domains. Calc. Var. Partial Differ. Equ (to appear).
[5] Kajikiya, R.: Partially symmetric solutions of the generalized Hénon equation in symmetric domains. Topol. Methods Nonlinear Anal (to appear). MR 1980135
[6] Kristály, A.: Asymptotically critical problems on higher-dimensional spheres. Discrete Contin. Dyn. Syst. 23 (2009), 919-935. DOI 10.3934/dcds.2009.23.919 | MR 2461832 | Zbl 1154.35051
[7] Lindqvist, P.: On the equation $ div (|\nabla u|^{p-2}\nabla u) + \lambda |u|^{p-2}u =0$. Proc. Am. Math. Soc. 109 (1990), 157-164. MR 1007505
[8] Montgomery, D., Samelson, H.: Transformation groups of spheres. Ann. Math. (2) 44 (1943), 454-470. DOI 10.2307/1968975 | MR 0008817 | Zbl 0063.04077
[9] Pontryagin, L. S.: Topological Groups. Transl. from the Russian, (3rd ed.), Classics of Soviet Mathematics Gordon and Breach, New York (1986). MR 0898007 | Zbl 0882.01025
Partner of
EuDML logo